Problem: A line whose $y$-intercept is $(0,5)$ intersects the ellipse $9x^2 + 16y^2 = 144.$  Find all possible slopes of this line.
Answer: The line is of the form $y = mx + 5.$  Substituting, we get
\[9x^2 + 16(mx + 5)^2 = 144.\]Expanding, we get
\[(16m^2 + 9) x^2 + 160mx + 256 = 0.\]For the line and ellipse to intersect, this quadratic must have a real root, which means that its discriminant is nonnegative:
\[(160m)^2 - 4(16m^2 + 9)(256) \ge 0.\]This reduces to $m^2 \ge 1.$  Thus, the possible slopes are $m \in \boxed{(-\infty,-1] \cup [1,\infty)}.$